Abstract not found

This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials f c (z) = z 2 + c, c 2 [\Gamma1; 1=2]; and (c) the family of rational maps f t (z) = z=t + 1=z, t 2 (0; 1]. We also calculate H: dim() ß 1:305688 for the Apollonian gasket, and H: dim(J(f)) ß 1:3934 for Douady's rabbit, where f(z) = z 2 + c satisfies f 3 (0) = 0. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Markov partitions and the eigenvalue algorithm . . . . . . . . 5 3 Schottky groups . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Markov partitions for polynomials . . . . . . . . . . . . ....

We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.

This paper investigates the behavior of the Hausdorff dimensions of the limit sets n and of a sequence of Kleinian groups \Gamma n ! \Gamma, where M = H 3 =\Gamma is geometrically finite. We show if \Gamma n ! \Gamma strongly, then: (a) Mn = H 3 =\Gamma n is geometrically finite for all n AE 0, (b) n ! in the Hausdorff topology, and (c) H: dim( n ) ! H: dim(), if H: dim() 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H: dim() ! 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer nc so that ∑∞ n=1 |(F n)′(F nc (c)) | −α < ∞ and F has no parabolic periodic cycles. Let µmax be the maximal multiplicity of the critical points. The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent α < exists a unique, ergodic, and non-atomic conformal measure ν with exponent δPoin(J) = HDim(J). If F is polynomially summable with the exponent α, ∑∞ n=1 n|(F n)′(F nc (c)) | −α < ∞ and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.

Abstract. We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the “hairiness phenomenon”, there exist many Feigenbaum Julia sets J(f) whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent δcr is equal to the hyperbolic dimension HDhyp(J(f)). Moreover, if area J(f) = 0 then HDhyp(J(f)) = HD(J(f)). In the stationary case, the last statement can be reversed: if area J(f)> 0 then HDhyp(J(f)) < 2. We also give a new construction of conformal measures on J(f) that implies that they exist for any δ ∈ [δcr, ∞), and analyze their scaling and dissipativity/conservativity properties.

Abstract. We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential −HD(J(f)) ln |f ′ |. 1.

Abstract. This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp(CK|s | δ) in strips |Re s | ≤ K, where δ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott-Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions {|Re s | ≤ |Im s | α} is given, followed by weaker lower bound estimates in strips {Re s> −C, |Im s | ≤ r}, and logarithmic neighbourhoods {|Re s | ≤ ρ log |Im s|}. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal. 1.

We introduce and establish some basic properties of the tame rational functions. The class of these functions contains all the rational functions with no recurrent critical points in their Julia sets. For tame non-exceptional functions we prove that the Lipschitz conjugacy, the same spectra of moduli of derivatives at periodic orbits and conformal conjugacy are mutually equivalent. We prove also the following rigidity result: If h is a Borel measurable invertible map which conjugates two tame functions f and g a.e. and if h transports conformal measure m f to a measure equivalent to m g then h extends from a set of full measure m f to a conformal homeomorphism of neighbourhoods of respective Julia sets. This extends D. Sullivan's rigidity theorem for holomorphic expanding repellers. We provide also a few lines proof of E. Prado's theorem that two generalized polynomial-like maps at zero Teichmüller's distance are holomorphically conjugate.

Because of its double periodicity, each elliptic function canonically induces a holomorphic dynamical system on a punctured torus. We introduce on this torus a class of summable potentials. With each such potential associated is the corresponding transfer (Perron-Frobenius-Ruelle) operator. The existence and uniquenss of ”Gibbs states” and equilibrium states of these potentials are proved. This is done by a careful analysis of the transfer operator which requires a good control of all inverse branches. As an application a version of Bowen’s formula for expanding elliptic maps is obtained.